41-comma: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 36 893 488 147 419 103 232 / <br>36 472 996 377 170 786 403 | | Ratio = 36 893 488 147 419 103 232 / <br>36 472 996 377 170 786 403 | ||
| Monzo = 65 -41 | | Monzo = 65 -41 | ||
| Name = 41-comma, Pythagorean countercomma, countercomp comma | |||
| Name = 41-comma, | | Color name = 41wM, fowowama | ||
| Color name = | | Comma = yes | ||
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}} | }} | ||
The '''41-comma''', '''Pythagorean countercomma''', or '''countercomp comma''' ([[monzo]]: {{monzo| 65 -41 }}), is a [[3-limit]] interval of 19.845 [[cent]]s. It is the amount by which a stack of 41 [[3/2|perfect fifths (3/2)]] falls short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup>. It can also be described as the difference between a perfect fifth and a stack of six [[2187/2048|apotomes]], or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth. | |||
== Temperaments == | == Temperaments == | ||
Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41 | Tempering out this comma leads to the [[countercomp]] temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]]. See [[countercomp family]] for a number of rank-2 temperaments where it is tempered out. | ||
[[ | |||
[[ | == Terminology == | ||
The names were given by [[Flora Canou]] in 2022. ''Pythagorean countercomma'' was derived by analogy to ''Pythagorean comma'' as part of a series of Pythagorean interval names. ''Countercomp comma'' was derived from the temperament name, ''countercomp'', which was changed from ''counterpyth'' as in earlier materials, where this comma was also called ''counterpyth comma''. It was renamed after the convention was established that the temperament of the [[Pythagorean comma]] should be [[compton]] and never ''Pythagorean'', for fear of confusion with [[Pythagorean tuning]]. | |||
== See also == | == See also == | ||
* [[Small comma]] | * [[Small comma]] | ||
* [[Schismic–countercommatic equivalence continuum]] | |||
[[Category: | [[Category:Countercomp]] | ||
[[Category: | [[Category:Commas named systematically]] | ||
[[Category: | [[Category:Commas named for their regular temperament properties]] | ||
Latest revision as of 14:46, 1 May 2026
| Interval information |
36 472 996 377 170 786 403
Pythagorean countercomma,
countercomp comma
reduced subharmonic
The 41-comma, Pythagorean countercomma, or countercomp comma (monzo: [65 -41⟩), is a 3-limit interval of 19.845 cents. It is the amount by which a stack of 41 perfect fifths (3/2) falls short of 24 octaves, in other words 224/(3/2)41. It can also be described as the difference between a perfect fifth and a stack of six apotomes, or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth.
Temperaments
Tempering out this comma leads to the countercomp temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions N up to 1230, the comma is tempered out if and only if 41 divides N. Examples are 41edo, 164edo, 205edo, 246edo, 328edo and 369edo. See countercomp family for a number of rank-2 temperaments where it is tempered out.
Terminology
The names were given by Flora Canou in 2022. Pythagorean countercomma was derived by analogy to Pythagorean comma as part of a series of Pythagorean interval names. Countercomp comma was derived from the temperament name, countercomp, which was changed from counterpyth as in earlier materials, where this comma was also called counterpyth comma. It was renamed after the convention was established that the temperament of the Pythagorean comma should be compton and never Pythagorean, for fear of confusion with Pythagorean tuning.